| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 28224.2-a1 |
28224.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{14} \cdot 7^{6} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$0.902969649$ |
1.089022372 |
\( \frac{267434754560}{546852789} a + \frac{622627300096}{546852789} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -30 a - 10\) , \( 60 a - 107\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-30a-10\right){x}+60a-107$ |
| 28224.2-b1 |
28224.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{14} \cdot 7^{6} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$0.902969649$ |
1.089022372 |
\( -\frac{267434754560}{546852789} a + \frac{296687351552}{182284263} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 30 a - 40\) , \( -60 a - 47\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(30a-40\right){x}-60a-47$ |
| 28224.2-c1 |
28224.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{6} \cdot 7^{8} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$6.070934530$ |
$1.390851865$ |
5.091785265 |
\( -\frac{2725888}{64827} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -7\) , \( 52\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-7{x}+52$ |
| 28224.2-c2 |
28224.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{20} \cdot 3^{24} \cdot 7^{2} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$6.070934530$ |
$0.347712966$ |
5.091785265 |
\( \frac{6522128932}{3720087} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -392\) , \( -228\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-392{x}-228$ |
| 28224.2-c3 |
28224.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{12} \cdot 7^{4} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$3.035467265$ |
$0.695425932$ |
5.091785265 |
\( \frac{6940769488}{35721} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -252\) , \( 1620\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-252{x}+1620$ |
| 28224.2-c4 |
28224.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{20} \cdot 3^{6} \cdot 7^{2} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$6.070934530$ |
$0.347712966$ |
5.091785265 |
\( \frac{7080974546692}{189} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -4032\) , \( 99900\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-4032{x}+99900$ |
| 28224.2-d1 |
28224.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{8} \cdot 7^{4} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.356039117$ |
$1.300573613$ |
3.350792649 |
\( -\frac{22638512}{11907} a + \frac{53307104}{11907} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 10 a - 37\) , \( -25 a + 85\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(10a-37\right){x}-25a+85$ |
| 28224.2-d2 |
28224.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{20} \cdot 3^{16} \cdot 7^{2} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.712078235$ |
$0.650286806$ |
3.350792649 |
\( \frac{2377183412}{413343} a + \frac{1866746596}{413343} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 10 a - 177\) , \( 87 a - 783\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(10a-177\right){x}+87a-783$ |
| 28224.2-e1 |
28224.2-e |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 7^{2} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.072121990$ |
$2.955494160$ |
7.198126740 |
\( -\frac{15893504}{15309} a + \frac{5612288}{5103} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -2 a - 1\) , \( -a - 2\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a-1\right){x}-a-2$ |
| 28224.2-f1 |
28224.2-f |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 7^{2} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 7 \) |
$0.072121990$ |
$2.955494160$ |
7.198126740 |
\( \frac{15893504}{15309} a + \frac{943360}{15309} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 4 a - 4\) , \( 4 a - 7\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(4a-4\right){x}+4a-7$ |
| 28224.2-g1 |
28224.2-g |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{8} \cdot 7^{4} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.356039117$ |
$1.300573613$ |
3.350792649 |
\( \frac{22638512}{11907} a + \frac{10222864}{3969} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -8 a - 28\) , \( 16 a + 32\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-8a-28\right){x}+16a+32$ |
| 28224.2-g2 |
28224.2-g |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{20} \cdot 3^{16} \cdot 7^{2} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.712078235$ |
$0.650286806$ |
3.350792649 |
\( -\frac{2377183412}{413343} a + \frac{1414643336}{137781} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -8 a - 168\) , \( -96 a - 864\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-8a-168\right){x}-96a-864$ |
| 28224.2-h1 |
28224.2-h |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{22} \cdot 3^{16} \cdot 7^{4} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 7 \) |
$1$ |
$0.282306202$ |
2.383318635 |
\( \frac{87211536659104}{234365481} a - \frac{125300355943606}{78121827} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -16 a + 2104\) , \( 22272 a - 9744\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-16a+2104\right){x}+22272a-9744$ |
| 28224.2-h2 |
28224.2-h |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{20} \cdot 3^{8} \cdot 7^{8} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$0.564612404$ |
2.383318635 |
\( \frac{3348718208}{5250987} a + \frac{1221326788}{1750329} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -16 a + 144\) , \( 320 a - 336\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-16a+144\right){x}+320a-336$ |
| 28224.2-i1 |
28224.2-i |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{18} \cdot 7^{2} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 7 \cdot 11 \) |
$0.019336474$ |
$1.160878407$ |
8.338316844 |
\( \frac{1700730880}{1240029} a - \frac{828596224}{413343} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -20 a - 7\) , \( 55 a - 61\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-20a-7\right){x}+55a-61$ |
| 28224.2-j1 |
28224.2-j |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{18} \cdot 7^{2} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 7 \cdot 11 \) |
$0.019336474$ |
$1.160878407$ |
8.338316844 |
\( -\frac{1700730880}{1240029} a - \frac{785057792}{1240029} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 20 a - 27\) , \( -55 a - 6\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(20a-27\right){x}-55a-6$ |
| 28224.2-k1 |
28224.2-k |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$4.843453083$ |
5.841424206 |
\( \frac{26112}{7} a + \frac{14080}{21} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -a + 3\) , \( -a\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-a+3\right){x}-a$ |
| 28224.2-l1 |
28224.2-l |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{22} \cdot 3^{16} \cdot 7^{4} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 7 \) |
$1$ |
$0.282306202$ |
2.383318635 |
\( -\frac{87211536659104}{234365481} a - \frac{288689531171714}{234365481} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 16 a + 2088\) , \( -22272 a + 12528\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(16a+2088\right){x}-22272a+12528$ |
| 28224.2-l2 |
28224.2-l |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{20} \cdot 3^{8} \cdot 7^{8} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$0.564612404$ |
2.383318635 |
\( -\frac{3348718208}{5250987} a + \frac{7012698572}{5250987} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 16 a + 128\) , \( -320 a - 16\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(16a+128\right){x}-320a-16$ |
| 28224.2-m1 |
28224.2-m |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{20} \cdot 3^{2} \cdot 7^{8} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$4.570174833$ |
$0.986178417$ |
10.87131181 |
\( \frac{11696828}{7203} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 48\) , \( 48\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+48{x}+48$ |
| 28224.2-m2 |
28224.2-m |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{16} \cdot 3^{4} \cdot 7^{4} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$2.285087416$ |
$1.972356834$ |
10.87131181 |
\( \frac{810448}{441} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -12\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-12{x}$ |
| 28224.2-m3 |
28224.2-m |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$4.570174833$ |
$3.944713669$ |
10.87131181 |
\( \frac{2725888}{21} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -7\) , \( -10\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-7{x}-10$ |
| 28224.2-m4 |
28224.2-m |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{20} \cdot 3^{8} \cdot 7^{2} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$4.570174833$ |
$0.986178417$ |
10.87131181 |
\( \frac{381775972}{567} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -152\) , \( 672\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-152{x}+672$ |
| 28224.2-n1 |
28224.2-n |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
28224.2 |
\( 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
$3.84140$ |
$(-a), (a-1), (2), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$4.843453083$ |
5.841424206 |
\( -\frac{26112}{7} a + \frac{92416}{21} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( a + 2\) , \( a - 1\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(a+2\right){x}+a-1$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
